I created a new index in my project to have two independent and different index, but the two index are connected. When I deleted one word in the new index, the same word is deleted in the first index.

When I've created the second index, I didn't choose the option "from an existing index", because I wanted a new and empty index but RH created a copy of the existing index and linked with the existing index.

It is possible toc reate a lot of different table of content and glossary but not different index ?

If you click File > Project Settings > General tab, is the index set to have keywords in Topics or the Index file (HHK)?

If it's set for Index file (HHK) that would likely explain it because the keywords are stored directly inside the topics themselves.

If the setting is already for Index file (HHK) I'm a bit stumped and can only fathom that maybe you have stumbled across a bit of a RoboHelp bug. I've not tried using multiple indexes before. Perhaps I should try and see.